The formation of extreme wave events, so-called rogue waves, can be understood and modelled by the nonlinear Schrödinger equation (NLSE) or its higher-order forms [1]}, [2]}, [3]}, [4]}. One way to control such rogue waves in a laboratory environment is the use of fundamental breather solutions in form of time-periodic, doubly-localized, and space-periodic forms. These are also known as Akhmediev (AB) [5]}, Peregrine (PB) [6]}, and Kuznetsov breather (KB) [7]}, respectively. Such solutions are based on long wave perturbation of a regular wave train with a precise dedicated phase-shift of the carrier at a specific location and instant.
While, modulational instability (MI) is usually seen as a mechanism that leads, through the amplification of sidebands, to a strong amplitude modulation, these phase-shifts always accompany such MI-induced amplitude reshaping.
Indeed, the initial temporal phase profile, or equivalently, the input phase relationship in Fourier space (i.e. the relative phase between the MI side-bands and the central carrier) have deep impact on the type of amplitude modulation
that develops upon propagation [8]}, [9]}, as recently observed experimentally [10]}, [11]}, [12]}.
